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Over thirty years ago, Kalai proved a beautiful $d$-dimensional analog ofCayley's formula for the number of $n$-vertex trees. He enumerated$d$-dimensional hypertrees weighted by the squared size of their$(d-1)$-dimensional homology group. This, however, does not answer the morebasic problem of unweighted enumeration of $d$-hypertrees, which is our concernhere. Our main result, Theorem 1.4, significantly improves the lower bound forthe number of $d$-hypertrees. In addition, we study a random $1$-out model of$d$-complexes where every $(d-1)$-dimensional face selects a random $d$-facecontaining it, and show it has a negligible $d$-dimensional homology.

Enumeration and randomized constructions of hypertrees